MTH 361 Numerical Analysis

%Calculate, by hand, the integral ^1 _0 ex dx using the midpoint, trapezoidal, and Simpson’s rules and
%3 subintervals. Use the error formula provided in class to estimate the minimum number m
%of subintervals that are needed for computing I(f) up to an absolute error ≤ 5 · 10−4 using the
%composite trapezoidal rule. Evaluate the absolute error that is actually made by coding it up.

% All of these functions must be saved as separate files in order to work.

% The Mid-Point Method
function ft = Method(f,a,b,n)

h = (b-a)/n;

S =0;

x = a+0.5*h;

for i =1:n

S =S+f(x);

x = x+h;

end

ft = h*S;

end

%Simposons_3 rule

function s = Simpson_3(f, a, b, n)

% The sample vector will be...

h = (b-a)./n;

xi = a:h:b;



fa = f(xi(1));

fb = f(xi(end));

% The even terms like f(x2), f(x4), ect...

feven = f(xi(3:2:end-2));

% Similarly, the odd terms like f(x1), f(x3), etc...

fodd = f(xi(2:2:end));

% Bringing everything together.

s = h / 3 * (fa + 2 * sum(feven) + 4 * sum(fodd) + fb);

end

% Trapezoidal Rule

function integral = Trap(a,b,n,f)

% f= Defined function using symstems.

% a= Initial point of integral.

% b= Last point of the interval.

% n= Number of sub-intervals must be the integer.

% Examples:

% syms x

% fun = x^2+x+1

% a = 0

% b = 1.6180

% n = 100

% result = trap(a, b, n, fun)



result=0;

f= inline(f);

h = (b-a)/n; %Finding the space between each subintervals.

x = [a+h:h:b-h]; %Finding the mid-points of each subintervals.

for i=1:n-1

result =result+f(x(i));

end

result=h*(result+0.5*(f(a)+f(b)));

integral=result;

end
% Inputs in the command window.
% f=@(x) exp(x);
% Method(f,0,1,8)
% Simpson_3(f,0,1,8)
% Trap(0,1,8,f)